The topology of knots and links in nematics

We review some our results concerning the topology of knotted and linked defects in nematic liquid crystals. We discuss the global topological classification of nematic textures with defects, showing how knotted and linked defect lines have a finite number of 'internal states', counted by the Alexander polynomial of the knot or link. We then give interpretations of these states in terms of umbilic lines, which we also introduce, as well as planar textures. We show how Milnor polynomials can be used to give explicit constructions of these textures. Finally, we discuss some open problems raised by this work.